There is an Archive option for GMail (default setup) and iCloud. When turned on they replace the trash can with a file cabinet to make it clear that the emails are not being deleted.
In GMail you can find the feature under Settings > Mail, Contacts, Calendars > GMail (your account). It is called Archive Messages. In my experience you end up with a coy of your email in the All Mail folder no matter what you do. I once fiddled around with some options on Google's webmail site and managed to turned it off, but not without turning off other desirable features as well. I gave up.
At any rate, turning it off will give you back the trash can and at least the illusion of deleting your email. And who knows, with the recent changes, maybe it even works as advertised now. It's been quite a while since I last looked.
The iCloud setting is deeper. You need to go to Settings > iCloud > Account > Mail (under Advanced). The Archive Messages switch is towards the bottom under Outgoing Mail Server. You can get to the same settings under in the Mail, Contacts, Calendars settings, using similar navigation.
It is possible some other accounts types have this option as well; these are the two that I know of.